How to find a finite splitting field $K$ for $G$ such that every indecomposable $KG$-module is absolutely indecomposable

Question

Let $G$ be a finite group and let $k$ be a finite field with char$(k)=p$ such that $p\mid |G|$.

If $k$ is a splitting field for $G$, then, no matter which splitting field we take, after extending scalars we always have the same number of isomorphism classes of simple modules.

However, it is not necessarily true that every indecomposable $kG$-module is absolutely indecomposable.

This leads to the following

Question: Is there a general construction that gives us a finite splitting field $K$ for $G$ (and all subgroups of all factor groups of $G$) such that every indecomposable $KG$-module (and every indecomposable K[H/J]-module where H/J is a factor group of a subgroup of G) is absolutely indecomposable?

Is there a reference in the literature?

Thanks for the help.

Answer 1

No.

Let $G=C_2\times C_2$, generated by elements $g$ and $h$, and let $K$ be any finite field of characteristic $2$.

Let $V$ be a finite dimensional $K$-vector space of dimension greater than one with an endomorphism $\varphi$ with irreducible characteristic polynomial $p(t)$.

Then $V\oplus V$ can be made into a $KG$-module by letting $g$ act as $\pmatrix{\text{id}_V & \text{id}_V\\0 & \text{id}_V}$ and $h$ act as $\pmatrix{\text{id}_V & \varphi\\0 & \text{id}_V}$, and it is indecomposable as a $KG$-module, but if $\overline{K}$ is an algebraic closure of $K$ then $V\otimes_K\overline{K}$ is a direct sum of two-dimensional $\overline{K}G$-modules corresponding to the roots of $p(t)$ in $\overline{K}$.